Scattering of kinks of the sinh-deformed ϕ4 model

Dionisio Bazeia,a Ekaterina Belendryasova,b Vakhid A. Gani,b,c,1

aDepartamento de Fısica, Universidade Federal da Paraıba, 58051-900 Joao Pessoa, Paraıba, BrazilbNational Research Nuclear University MEPhI

(Moscow Engineering Physics Institute), 115409 Moscow, RussiacTheory Department, National Research Center Kurchatov Institute, Institute for Theoretical and

Experimental Physics, 117218 Moscow, Russia

E-mail: bazeia@fisica.ufpb.br, vagani@mephi.ru

Abstract: We consider the scattering of kinks of the sinh-deformed ϕ4 model. This

model is obtained from the well-known ϕ4 model by means of the deformation procedure.

Depending on the initial velocity of the colliding kinks, we observe different regimes of

the collision. There is a critical value vcr of the initial velocity, which separates reflection

(at vin > vcr) and complicated interaction (at vin < vcr) with kinks’ capture and escape

windows. Besides that, below vcr we observe formation of a bound state of two oscillons,

as well as escape of oscillons for some values of the initial velocity of the colliding kinks.

1Corresponding author.

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Contents

1 Introduction 1

2 Topological solitons in (1,1)-dimensional models 3

3 The ϕ4 model 6

4 Deformation procedure and the sinh-deformed ϕ4 model 8

5 Kink-antikink collisions in the sinh-deformed ϕ4 model 10

6 Comments and Conclusion 13

1 Introduction

Topological defects arise in a diversity of contexts in high energy physics, cosmology, quan-

tum and classical field theory, condensed matter, and so on. In high energy physics, they are

topologically non-trivial solutions of the equations of motion and possess very interesting

properties, which lead to new physical phenomena [1–4].

Nowadays, the study of the topological defects is a very fast developing area with

significant progress being achieved in the investigation of domain walls, vortices, strings,

as well as embedded topological defects such as Q-lump on a domain wall and skyrmion on a

domain wall, and so on [5–14]. Besides, it is also of interest to mention the so-called Q-balls

and similar configurations [15–20], which are charged and then protected against decaying

into the elementary excitations supported by the model under consideration. Also, it is

worth mentioning other possibilities, such as the study of solitons in fibers [21], bubble

collisions in cosmology [22], and localized excitations in nonlinear systems [23].

The (1, 1)-dimensional models are of special interest [2, 3, 24], since the dynamics of

some two- or three-dimensional systems can be reduced to the one-dimensional models.

For example, a planar domain wall, which separates regions with different minima of the

potential, in the direction perpendicular to it can be interpreted as a one-dimensional topo-

logical configuration (a kink). On the other hand, the (1, 1)-dimensional field-theoretical

models can be a first step toward more complicated higher dimensional models. Moreover,

even in the (1, 1)-dimensional case, topological defects may arise in the more complex mod-

els with two or more fields; see, e.g., refs. [25–51]. In this more general context, several

works have been developed to find analytical solutions, to study their stability and to use

them in application of interest in physics [25–36]. Other investigations have dealt with

the presence of junctions and/or intersections of defects [37–40], and with issues related to

composite-kink internal structures, twinlike models with several fields and triplet scalar on

domain walls [41–51], among other issues.

– 1 –

In the case of models described by real scalar fields with standard kinematics, in the

(1, 1) space-time the presence of interactions that develop spontaneous symmetry breaking

in general leads to localized topological structures having the kinklike profile. The inter-

actions of these one-dimensional topological structures with each other and with spatial

inhomogeneities (impurities) have attracted the attention of physicists and mathematicians

during a long time; see, e.g., ref. [24]. The first studies on this subject date back to the

1970s and 1980s [52, 53]. Nevertheless, forty years later we see that it is still an actively

developing area with many new applications. Many important results have been obtained

by means of the numerical simulation, which is one of the most powerful tools for studying

the subject. In particular, resonance phenomena – escape windows and quasi-resonances –

were found in the kink-antikink scattering process. A broad class of the (1, 1)-dimensional

models with polynomial potentials such as the ϕ4, ϕ6, ϕ8, and others with polynomial

self-interaction with higher degree has been considered [53–66]. Also, one should mention

the new results on the long-range interaction between kinks [65–69]. Other models with

non-polynomial potentials are also being discussed in the literature. For example, the mod-

ified sine-Gordon [70], the double sine-Gordon [71–73], and a variety of models which can

be obtained using the deformation procedure, which we explain below.

Apart from the numerical solving of the equation of motion, other methods are widely

used for investigating the kink-antikink interactions. One of them is the collective co-

ordinate method [55, 74–80]. Within this approximation a real field-theoretical system

(which formally has an infinite number of degrees of freedom) is approximately described

as a system with one or several degrees of freedom. For example, in the case of the kink-

antikink configuration one can use the distance between the kink and the antikink as the

only degree of freedom (collective coordinate). In more complicated modifications of this

approximation some other degrees of freedom (for instance, vibrational) can be involved,

see, e.g., [74–76]. Another approximation, which allows to estimate the force between kink

and antikink, is Manton’s method [3, Ch. 5], [81–84]. This method is based on using the

kinks’ asymptotics in situations where the distance between kinks is large. However, one

should mention that the applicability of this method for kinks and solitons with power-law

asymptotics is not obvious.

An impressive progress has been achieved in the analytical treatment of the (1, 1)-

dimensional field-theoretical models. Among several possibilities to deal with the problem

analytically, the trial orbit method was suggested in [25] as a way to solve the equations of

motion which appear in systems described by two real scalar fields that interact nonlinearly.

This method has been used by others, and in [43] it was shown to be very effective when

the equations of motion can be reduced to first-order differential equations. Also, in [44]

the authors have used the integrating factor to solve the equations of motion in the case

of a very specific potential.

Another possibility of searching for models that support analytical solutions appeared

before in [85] and also in refs. [86–88]. It refers to the deformation procedure, a method

of current interest which helps us to introduce new models, and solve them analytically.

This will be further reviewed below, and used to define the model we want to investigate

in the current work. In particular, the new model is somehow similar to the ϕ4 model with

– 2 –

spontaneous symmetry breaking, so we will compare the current investigation with the ϕ4

case, to highlight the differences between the two cases, and to see how the non-polynomial

interaction that appears in the new model contributes to modify the behavior which is

present in the standard ϕ4 model.

In this work we focus our attention on the kink-antikink scattering process and orga-

nize the investigation as follows. In section 2 we give general introduction to the (1, 1)-

dimensional field-theoretical models, which possess topological solutions with the kink pro-

file. In section 3 we review the ϕ4 model, briefly accounting for the kink-antikink scattering

within this model. Furthermore, in section 4 we apply the deformation procedure to the

ϕ4 model in order to introduce a model with non-polynomial potential, which we call the

sinh-deformed ϕ4 model. In section 5 we focus on the collisions of the kink and the antikink

of the sinh-deformed ϕ4 model. In this section we present our main results and compare

them with the results of the ϕ4 model. Finally, in section 6 we conclude with a discussion

of the results and the prospects for future works.

2 Topological solitons in (1,1)-dimensional models

Consider a field-theoretical model in the (1, 1)-dimensional space-time with its dynamics

defined by the Lagrangian

L =1

2

(∂ϕ

∂t

)2

− 1

2

(∂ϕ

∂x

)2

− U(ϕ), (2.1)

where ϕ(x, t) is a real scalar field. The potential U(ϕ) is supposed to be non-negative func-

tion with two or more degenerate minima, ϕ(0)1 , ϕ

(0)2 , . . . , such that U(ϕ

(0)1 ) = U(ϕ

(0)2 ) =

... = 0. The Lagrangian (2.1) leads to the following equation of motion for the field ϕ

∂2ϕ

∂t2− ∂2ϕ

∂x2+dU

dϕ= 0. (2.2)

The energy functional corresponding to the Lagrangian (2.1) is

E[ϕ] =

∫ ∞−∞

[1

2

(∂ϕ

∂t

)2

+1

2

(∂ϕ

∂x

)2

+ U(ϕ)

]dx. (2.3)

In the static case∂ϕ

∂t= 0, and from eq. (2.2) we have

d2ϕ

dx2=dU

dϕ. (2.4)

This equation can be easily transformed into the first order differential equations

dϕ

dx= ±√

2U. (2.5)

For the energy of the static configuration to be finite, the two following conditions must

hold

limx→−∞

ϕ(x) = ϕ(0)i and lim

x→+∞ϕ(x) = ϕ

(0)j , (2.6)

– 3 –

where ϕ(0)i and ϕ

(0)j are two adjacent minima of the potential. These expressions (2.6) are

necessary conditions for the energy of a static configuration to be finite. If (2.6) hold, then

the second and the third terms in the integrand in (2.3) fall off at x → ±∞, hence the

integral (2.3) can be convergent. Configurations with ϕ(0)i 6= ϕ

(0)j are called topological and

have a kinklike shape. In this sense, a conserved topological current can be introduced,

and for the models to be investigated below one can use

jµtop =1

2εµν∂νϕ. (2.7)

The corresponding topological charge is

Qtop =

∫ ∞−∞

j0topdx =

1

2[ϕ(+∞)− ϕ(−∞)] . (2.8)

This charge is determined only by the asymptotics (2.6), so it does not depend on the

behavior of the field ϕ(x) at finite x.

For every non-negative potential U(ϕ) we can introduce a smooth function W (ϕ),

called the superpotential, as

U(ϕ) =1

2

(dW

dϕ

)2

. (2.9)

Using the superpotential we can rewrite the energy of a static configuration ϕ(x) in the

following manner

E = EBPS +1

2

∫ ∞−∞

(dϕ

dx± dW

dϕ

)2

dx, (2.10)

where

EBPS = |W [ϕ(+∞)]−W [ϕ(−∞)]|. (2.11)

From eq. (2.10) one can see that the energy of any static configuration belonging to a given

topological sector is bounded from below by EBPS. The configurations with the minimal

energy (2.11) are called BPS configurations, or BPS saturated configurations [89, 90]. From

eq. (2.10) it is easy to see that any BPS configuration satisfies the first order differential

equationsdϕ

dx= ±dW

dϕ, (2.12)

which coincide with (2.5).

Below we deal with kinks and antikinks — the BPS saturated topological solutions

of eq. (2.5), which interpolate between neighboring minima of the potential. The solution

with the asymptitics ϕ(+∞) > ϕ(−∞) is called kink, while the term antikink stands for

the solution with ϕ(+∞) < ϕ(−∞). Sometimes we use the term kink for both kink and

antikink, for brevity.

Many phenomena observed in the kink-antikink scattering can be explained by the

presence of the vibrational mode(s) in the kink’s excitation spectrum. In order to find the

spectrum of localized excitations of a kink, we have to add a small perturbation δϕ(x, t)

to the static kink solution ϕk(x),

ϕ(x, t) = ϕk(x) + δϕ(x, t), |δϕ| � |ϕk|. (2.13)

– 4 –

The substitution of ϕ(x, t) into the equation of motion (2.2) leads to the partial differential

equation for the perturbation δϕ(x, t); after linearization one gets

∂2δϕ

∂t2− ∂2δϕ

∂x2+d2U

dϕ2

∣∣∣∣ϕk(x)

δϕ = 0. (2.14)

Since the second derivative of the potential calculated at the static solution ϕk(x) depends

only on x, we can assume that δϕ has the form

δϕ(x, t) = η(x) cos ωt, (2.15)

and this allows us to obtain the eigenvalue problem of the type of the stationary Schrodinger

equation,

Hη(x) = ω2η(x), (2.16)

where the operator H (the Hamiltonian) is

H = − d2

dx2+ u(x), (2.17)

with the potential

u(x) =d2U

dϕ2

∣∣∣∣ϕk(x)

. (2.18)

For each state of the discrete spectrum, the corresponding eigenfunction η(x) is a twice

continuously differentiable and square-integrable on the x-axis. Kink and antikink have

the same excitation spectrum.

The discrete spectrum in the potential (2.18) always possesses a zero (or translational)

mode ω0 = 0. It can easily be shown by differentiating eq. (2.4) with respect to x, and

taking into account that ϕk(x) is a solution of eq. (2.4), i.e.

− d2

dx2

dϕk

dx+d2U

dϕ2

∣∣∣∣ϕk(x)

dϕk

dx= 0, or H ·

dϕk

dx= 0. (2.19)

So we see thatdϕk

dxis an eigenfunction of the Hamiltonian (2.17) associated with the

eigenvalue ω0 = 0. The presence of a zero mode in the kink’s excitation spectrum is a

consequence of the translational invariance of the Lagrangian.

Furthermore, the presence of W in eq. (2.9) allows to write the operator H in the form

H = A†A, (2.20)

where A† and A are the first order differential operators

A† = − d

dx+d2W

dϕ2

∣∣∣∣ϕk(x)

, A =d

dx+d2W

dϕ2

∣∣∣∣ϕk(x)

. (2.21)

This factorization shows that the operator H is non-negative, so the static solution ϕk(x)

is linearly stable.

– 5 –

3 The ϕ4 model

In this section we recall some facts about kinks of the ϕ4 model. We use for simplicity

dimensionless fields and space-time coordinates and write the potential of the ϕ4 model in

the form

U1(ϕ) =1

2(1− ϕ2)2. (3.1)

This potential possesses two degenerate minima ϕ(0)1 = −1 and ϕ

(0)2 = 1, see figure 1(a).

The equation (2.5) with the potential (3.1) can be easily integrated, which yields the static

-1.5 -1.0 -0.5 0.5 1.0 1.5 φ

0.2

0.4

0.6

0.8

1.0

U

(a) Potentials

-4 -2 2 4 x

-1.0

-0.5

0.5

1.0

φ

(b) Kinks

Figure 1. Potentials and kinks of the ϕ4 model (dashed curves) and of the sinh-deformed ϕ4 model

(solid curves).

topologically non-trivial solutions, the kink and the antikink:

ϕk(x) = tanhx, ϕk(x) = − tanhx. (3.2)

These kinks interpolate between the two minima of the potential, as shown in figure 1(b).

The mass of the kink (antikink), i.e. the energy E[ϕk(x)] of the static kink (antikink), is

Mk =4

3. (3.3)

The moving kink (antikink) can be obtained from eq. (3.2) by the Lorentz boost.

The quantum-mechanical potential (2.18), which defines the spectrum of the localized

excitations of the ϕ4 kink, has the form

u1(x) = 4− 6

cosh2 x. (3.4)

It is the well-known modified Poschl-Teller potential [91]. Apart from the zero mode ω0 = 0,

there is a vibrational mode with the frequency ω1 =√

3. As we explain below, the presence

of the vibrational mode leads to resonance phenomena in the kink-antikink collisions.

As we informed in the Introduction, the scattering of the ϕ4 kinks is well-studied, so

let us now briefly review the main features of the collision processes in this case.

– 6 –

Consider the initial configuration in the form of kink and antikink centered at the

points x = −x0 and x = x0, respectively, and moving towards each other with the initial

velocities vin in the laboratory frame, i.e.

ϕ(x, t) = tanh

x+ x0 − vint√1− v2

in

− tanh

x− x0 + vint√1− v2

in

− 1. (3.5)

To find evolution of this initial configuration, we solved the equation of motion (2.2) with

the potential (3.1) numerically using the standard explicit finite difference scheme,

∂2ϕ

∂t2=ϕj+1i − 2ϕji + ϕj−1

i

δt2,

∂2ϕ

∂x2=ϕji+1 − 2φji + ϕji−1

δx2, (3.6)

where (i, j) number the x and t coordinates of the grid points, (xi, tj), on a grid with the

steps δt = 0.008 and δx = 0.01. We repeated selected computations with smaller steps,

δt = 0.004 and δx = 0.005, in order to check our numerical results. We also checked the

total energy conservation. In all simulations of the ϕ4 kinks collisions we used the initial

half-distance x0 = 5.

Depending on the initial velocity, the kinks scattering looks differently. There is a

critical value of the initial velocity vcr ≈ 0.2598. At vin > vcr we observe kinks escape after

a collision, see figure 2. Some part of the energy is being emitted in the form of small

10 20 30 40 50t

-1.5

-1.0

-0.5

0.5

φ(0,t)

Figure 2. Bounce off and escape of kinks of the ϕ4 model at vin = 0.4000. Left panel — the

space-time picture of the field evolution. Right panel — the time dependence of the field at the

origin.

waves.

At vin < vcr the kinks collide and form a long-living bound state, a bion, which is

illustrated in figure 3. This bion decays slowly, emitting its energy in the form of waves of

small amplitude. However the kinks capture appears not for all vin < vcr, since there is a

pattern of escape windows in the collision processes. An escape window refers to a narrow

interval of initial velocities, within which kinks do not form a bound state but escape to

infinities. It is important point that, unlike bouncing off at vin > vcr, within an escape

window the kinks escape to infinities after two, three or more collisions. According to the

number of collisions before escaping, there are two-bounce windows, three-bounce windows,

– 7 –

20 40 60 80 100 120 140t

-1.5

-1.0

-0.5

0.5

φ(0,t)

Figure 3. The formation of a bound state in the ϕ4 kink-antikink collision at vin = 0.2541. Left

panel — the space-time picture of a bion formation. Right panel — the time dependence of the

field at the origin.

and so on. See figure 4 for some illustrations of two-, three- and four-bounce windows. The

escape windows form a fractal structure. Two-bounce windows are the broadest, and near

each of them there is a series of three-bounce windows. Near each three-bounce window,

in turn, there is a series of four-bounce windows, and so on, see, e.g., [24].

The explanation of the appearance of the escape windows is that they are related to

the resonance energy exchange between the kinetic energy (the translational mode) and

the vibrational mode of the kink (antikink). The mechanism works as follows: consider,

for example, the two-bounce window illustrated in figure 4(a). At the first collision, some

part of the kinks kinetic energy is transferred into their vibrational modes. As a result of

the loss of the kinetic energy, the kink and the antikink are not able to overcome mutual

attraction, and they return and collide again. However, if a certain resonance relation

between the time T12 between the first and the second collisions and the frequency ω1 of

the vibrational mode holds, a part of the energy can be returned into the kinetic energy,

and the kinks are then able to escape from each other.

4 Deformation procedure and the sinh-deformed ϕ4 model

The sinh-deformed ϕ4 model can be obtained from the ϕ4 model by using the deformation

procedure [85–88, 92]. The potential U2(ϕ) of the new model is related with the old model

potential U1(ϕ) by a deforming function f(ϕ),

U2(ϕ) =U1(ϕ→ f(ϕ))

(df/dϕ)2, (4.1)

where “ϕ→ f(ϕ)” means that one must substitute the field ϕ by f(ϕ). At the same time,

the kink of the new model, ϕ(new)k (x), can be easily obtained from the kink of the old model,

ϕ(old)k (x), by the inverse deforming function f−1,

ϕ(new)k (x) = f−1(ϕ

(old)k (x)). (4.2)

– 8 –

20 40 60 80 100 120 140t

-1.5

-1.0

-0.5

0.5

φ(0,t)

(a) Two-bounce window, vin = 0.2528

20 40 60 80 100t

-1.5

-1.0

-0.5

0.5

φ(0,t)

(b) Three-bounce window, vin = 0.1916

20 40 60 80 100 120 140t

-1.5

-1.0

-0.5

0.5

φ(0,t)

(c) Four-bounce window, vin = 0.2504

Figure 4. Examples of two-, three-, and four-bounce windows in the collisions of the ϕ4 kinks.

Left panel — the space-time picture of the escape windows. Right panel — the time dependence

of the field at the origin.

We start from the ϕ4 model with the potential (3.1) and use the deforming function

f(ϕ) = sinhϕ. Then we come to the potential of the sinh-deformed ϕ4 model

U2(ϕ) =1

2sech2ϕ

(1− sinh2 ϕ

)2. (4.3)

This potential has two degenerate minima, ϕ± = ±arsinh 1, V (ϕ±) = 0, see figure 1(a).

– 9 –

The kinks of the sinh-deformed ϕ4 model are

ϕk(x) = arsinh(tanhx), ϕk(x) = −arsinh(tanhx), (4.4)

see figure 1(b). The mass of the sinh-deformed ϕ4 kink (antikink), i.e. the energy E[ϕk(x)]

of the static kink (antikink), is

Mk = π − 2. (4.5)

The excitation spectrum of the kink (antikink) (4.4) is defined by the quantum-

mechanical potential

u2(x) = 2 tanh2 x+ 1 +8 tanh2 x− 4

(1 + tanh2 x)2, (4.6)

which is presented in figure 5. We performed a numerical search of the discrete part

-3 -2 -1 1 2 x

-3

-2

-1

1

2

3

u

Figure 5. The quantum-mechanical potential (4.6).

of the excitation spectrum in the potential (4.6). This problem was solved using the

standard shooting method. For various values of ω2 we integrated eq. (2.16) with the

known asymptotic behavior η(x) ∼ exp(−√

4− ω2|x|) of its solutions at x→ ±∞, starting

from a large negative x and from a large positive x. As a result, we obtained two different

solutions, the “left” solution and the “right” solution, which were then matched at some

point x close to the origin (the particular choice of x is not important). The Wronskian of

the “left” and the “right” solutions, calculated at the matching point, as a function of ω2

turns to zero at eigenvalues of the Hamiltonian (2.17) with the potential (4.6).

We found two levels in the potential (4.6): the zero mode ω0 = 0, and the vibrational

mode with the frequency ω1 ≈ 1.89.

5 Kink-antikink collisions in the sinh-deformed ϕ4 model

We studied the collisions of the kink and the antikink of the sinh-deformed ϕ4 model

using the initial configuration similar to that used in section 3 in the case of the ϕ4 kinks

scattering, namely

ϕ(x, t) = arsinh

tanh

x+ x0 − vint√1− v2

in

− arsinh

tanh

x− x0 + vint√1− v2

in

− arsinh 1,

(5.1)

– 10 –

which corresponds to the kink and the antikink centered at x = ±x0 and moving towards

each other with the initial velocities vin. We used 2x0 = 10 and the same parameters of the

numerical scheme as for the ϕ4 kinks in section 3, see eq. (3.6) and the paragraph below

this equation.

We found a critical value of the initial velocity vcr ≈ 0.4639, which separates two

different regimes of the kinks scattering. At vin > vcr the kinks bounce off and escape to

infinities after one collision. This is illustrated in figure 6, and the situation here is similar

to that observed for the ϕ4 kinks above the critical velocity, as depicted in figure 2.

20 40 60 80t

-1.5

-1.0

-0.5

0.5

φ(0,t)

Figure 6. Bounce off and escape of kinks of the sinh-deformed ϕ4 model at vin = 0.5000. Left

panel — the space-time picture of the field evolution. Right panel — the time dependence of the

field at the origin.

At the initial velocities below the critical value, vin < vcr, we observed the kinks’

capture and formation of their bound state, figure 7, and a rich variety of resonance phe-

20 40 60 80t

-1.5

-1.0

-0.5

0.5

φ(0,t)

Figure 7. The formation of a bound state in the sinh-deformed ϕ4 kink-antikink collision at

vin = 0.4000. Left panel — the space-time picture of a bion formation. Right panel — the time

dependence of the field at the origin.

nomena. First of all, in this range of the initial velocities we found a complicated pattern

of escape windows, similar to the case of the ϕ4 kinks. We identified many two-bounce,

three-bounce, etc., escape windows. The field dynamics within these windows is similar

to the case of the ϕ4 model. In figure 8 we present examples of the field behavior within

– 11 –

20 40 60t

-1.5

-1.0

-0.5

0.5

φ(0,t)

(a) Two-bounce window, vin = 0.4600

20 40 60t

-1.5

-1.0

-0.5

0.5

φ(0,t)

(b) Three-bounce window, vin = 0.4500

Figure 8. Examples of two- and three-bounce windows in the collisions of kinks of the sinh-

deformed ϕ4 model. Left panel — the space-time picture of the escape windows. Right panel

— the time dependence of the field at the origin.

two- and three-bounce windows. In a way similar to the case of the ϕ4 model, the escape

windows that appear in the sinh-deformed model seem to form a fractal structure. We

found several three-bounce escape windows near a two-bounce window, see figure 9. Near

one of the three-bounce escape windows we observed four-bounce escape windows. This

behavior is similar to the one found in the ϕ4 model, so it suggests that the escape windows

also form a fractal structure in this case.

At the same time, the bion formation in the range vin < vcr outside the escape windows

looks differently. In our numerical experiments we observed new phenomena, which are

not typical for the ϕ4 kinks. In many cases the final configuration looked like a bound

state of two oscillons. These oscillons oscillate around each other near the origin and, as a

consequence, the dependence on time of the field at the origin has a low-frequency envelope,

as it is shown in figure 10. One notes that the amplitude and frequency of oscillations of

these structures depend on the initial velocity of the colliding kinks. Moreover, at some

values of the initial velocity we observed escape of the two oscillons with the final velocity

vos, which varies in a wide range, as one can see from figure 11. The situation can be

interpreted as follows: at some initial velocities of the colliding kinks the bion is formed,

which evolves rather fast into a bound state of two oscillons, which can either oscillate

– 12 –

Figure 9. Top panel — a two-bounce escape window and some of the nearby three-bounce

windows. Bottom panel — a three-bounce window (which is boxed on the top panel) with some

nearby four-bounce windows.

around each other, or escape to infinities. The intervals of the initial velocity of the colliding

kinks, at which the oscillons escape, form oscillons’ escape windows. The frequency of the

field oscillations is the same for all oscillons, ωos ≈ 1.88, which is very close to ω1 = 1.89.

In figure 12 we show the dependence of the period of oscillations on the initial velocity

of the colliding kinks. The shaded areas denote the escape windows for oscillons, i.e. the

intervals of the initial velocity, at which the two oscillons escape to infinities. The widths

of the escape windows are 0.00030, 0.00013, 0.00011, 0.00002, and ∼ 0.00001 for the 1st,

2nd, 3rd, 4th, and 5th windows, respectively.

The high frequency in figure 10 (right panels) is close to the frequency of the vibrational

mode ω1 ≈ 1.89 of the sinh-deformed ϕ4 kink. For example, at the initial velocity vin =

0.44183 the frequency is 1.86, at vin = 0.44188 it is 1.84, and at vin = 0.44190 it equals

1.83.

6 Comments and Conclusion

In this work, we investigated the scattering of kinks of the sinh-deformed ϕ4 model, which

can be obtained from the ϕ4 model by the deformation procedure. In fact, for pedagogical

reasons we firstly reviewed the scattering of kinks in the ϕ4 model and then studied the

same issue in the case of the sinh-deformed ϕ4 model. We noted that the two models

engender similar behavior in several aspects: they support similar kinklike configurations

and their stability potentials present almost the same profile, which gives rise to the zero

mode and another bound state, the vibrational state with the frequency ω1 =√

3 ≈ 1.73

in the case of the ϕ4 model, and ω1 ≈ 1.89 for the sinh-deformed ϕ4 model.

Moreover, in the scattering of kinks, the two models also admit a critical velocity vcr,

which separates two different regimes of the collisions. On the one hand, at vin < vcr we

observed the capture of kinks and the formation of bound states and, on the other hand,

– 13 –

50 100 150 200t

-1.5

-1.0

-0.5

0.5

φ(0,t)

(a) Small amplitude, vin = 0.44183

50 100 150 200t

-1.5

-1.0

-0.5

0.5

φ(0,t)

(b) Medium amplitude, vin = 0.44188

50 100 150 200t

-1.5

-1.0

-0.5

0.5

φ(0,t)

(c) Large amplitude, vin = 0.44190

Figure 10. Formation of a bound state of two oscillons in the collisions of kinks of the sinh-

deformed ϕ4 model. Left panel — the space-time picture. Right panel — the time dependence

of the field at the origin.

for vin > vcr the kinks escape to infinity after one collision. The value of the critical velocity

is vcr = 0.4639 for the sinh-deformed ϕ4 model and for the ϕ4 model it is equal to 0.2598.

In the study of collisions of kinks in the sinh-deformed ϕ4 model, we observed that

for velocities in the range vin < vcr, there appeared several escape windows, which are also

specific for the ϕ4 and some other models. We have found two-bounce, three-bounce and

other escape windows, and here we recall that within an n-bounce window the kinks escape

– 14 –

(a) vin = 0.44191, vos ≈ 0.04

(b) vin = 0.44195, vos ≈ 0.10

(c) vin = 0.44207, vos ≈ 0.18

(d) vin = 0.44220, vos ≈ 0.06

Figure 11. Escape of two oscillons for several values of the initial velocity of the colliding kinks

vin, here vos stands for the final velocity of the escaping oscillon.

Figure 12. The period of the oscillons’ oscillations about each other as a function of the initial

velocity of the colliding kinks. The blue-shaded areas denote the escape windows for oscillons. The

green-shaded area denotes the two-bounce escape window for kinks, while the pink-shaded areas

denote the four-bounce escape windows for kinks.

to infinities after n collisions. The cause of the escape windows is somehow related to the

resonant energy exchange between the translational and the vibrational modes of the kink

and the antikink.

– 15 –

The general results of the kink collisions which appear in the sinh-deformed model

suggest that the model is not integrable, and that its kinklike configuration is not a soliton.

This is of current interest, and we may make it stronger if one recalls the sine-Gordon model,

which is integrable model [93]. It is interesting to see that this model can also be obtained

from the ϕ4 model with the deformation procedure. We use the deformation function

f(ϕ) = sinϕ, the potential U1(ϕ) in eq. (3.1) and the procedure illustrated in eq. (4.1) to

get

U3(ϕ) =1

2cos2 ϕ. (6.1)

This is the potential of the sine-Gordon model, and its kinklike or better soliton solution

can be written in the form

ϕs(x) = arcsin(tanhx). (6.2)

As it is well-known, the corresponding stability potential supports the zero mode and no

other bound state, and this helps us to understand its integrability. The sine-Gordon

potential is periodic and so well-different from the ϕ4 model, which is described by a

polynomial potential. In this sense, the sinh-deformed model which we have studied in this

work seems to be farther away from the sine-Gordon and integrability, and hence it should

present information that is absent in the ϕ4 model.

With this motivation in mind, we then looked deeper into the escape windows in the

sinh-deformed model and observed a new phenomenon, the conversion of the kink-antikink

pair into a complex oscillating structure at the collision point at the origin. This structure

can be interpreted as a bound state of two individual oscillons. It is interesting that at

some initial velocities of the colliding kinks we observed escape of these two oscillons.

The interval of initial velocities of the kinks, in which the kinks collide and form a

bound state of two oscillons, which then escape, can be called an oscillons’ escape window.

In our simulations the final velocity of the escaping oscillons varies in a wide range from

zero to ∼ 0.2. Near the oscillons’ escape window the period of the oscillons’ oscillations in

their bound state increases, see figure 12.

As shown in the recent work [92], we can introduce other models using the deformation

function of the hyperbolic type. In particular, we can start with the ϕ6 model studied before

in [54], which supports no vibrational state. For instance, we can use the potential

U4(ϕ) =1

2ϕ2(1− ϕ2)2, (6.3)

and the deformation function f(ϕ) = sinhϕ to get to a new model

U5(ϕ) =1

2tanh2 ϕ (1− sinhϕ)2. (6.4)

We note that for ϕ very small, the above model (6.4) leads us back to the model in (6.3). We

call this model (6.4) the sinh-deformed ϕ6 model. An interesting issue would be to study

the scattering of kinks in this model, to see how it can be connected to the investigation

[54–57], which revealed a resonant scattering structure that provided a counterexample to

the belief that the existence of the vibrational bound state is a necessary condition for the

– 16 –

appearance of multibounce resonances. Another issue is the study of the force between

kinks, to see if it can be connected with the scattering of kinks.

We can also consider models with modified kinematics, as the ones recently investigated

in [94], where one considers the Dirac-Born-Infeld case. This modification changes the

standard scenario and may contribute to add new possibilities to the escape windows that

appear in the standard situation. Another route concerns models described by two real

scalar fields, as the one investigated in refs. [29, 41]. In this case, the presence of the two

fields leads to analytical kinklike solutions which have internal structure that can be used

to model Bloch walls. The scenario here is richer and an interesting issue would be the

study of the scattering of kinks in this model, to see how the internal structure contributes

to the presence of the escape windows, etc. These and other similar issues are currently

under consideration, and we hope to report on them in the near future.

Acknowledgments

This work was performed using resources of NRNU MEPhI high-performance comput-

ing center. The research was supported by the Brazilian agency CNPq under contracts

455931/2014-3 and 306614/2014-6, and by the MEPhI Academic Excellence Project under

contract No. 02.a03.21.0005, 27.08.2013.

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